We are concerned with scaling limits of the solutions to stochastic differential equations with stationary coefficients driven by Poisson random measures and Brownian motions. We state an annealed convergence theorem, in which the limit exhibits a diffusive or superdiffusive behavior, depending on the integrability properties of the Poisson random measure
We discuss in a stochastic framework the interplay between Riemann-Liouville type operators applied to stochastic processes, real interpolation, bounded mean oscillation, and an approximation problem for stochastic integrals. We provide upper and lower bounds for gradient processes on the Lévy-Itô space, which arise in the special case of the Wiener space from the Feynman-Kac theory for parabolic PDEs. The upper bounds are formulated in terms of BMO-conditions on the fractional integrated gradient, the lower bounds in terms of oscillatory quantities. On the general Lévy-Itô space we are concer…
A step reinforced random walk is a discrete time process with memory such that at each time step, with fixed probability $p \in (0,1)$, it repeats a previously performed step chosen uniformly at random while with complementary probability $1-p$, it performs an independent step with fixed law. In the continuum, the main result of Bertoin in [7] states that the random walk constructed from the discrete-time skeleton of a Lévy process for a time partition of mesh-size $1/n$ converges, as $n \uparrow \infty$ in the sense of finite dimensional distributions, to a process $\hatξ$ referred to as a no…
This a free translation with additional explanations of {\em Processus à Accroissement Independants Chapitre I: La Décomposition de Paul Lévy}, by J.L. Bretagnolle, in {\em Ecole d'Eté de Probabilités}, Lecture Notes in Mathematics 307, Springer 1973. The Lévy-Khintchine representation of infinitely divisible distributions is obtained as a by-product. As this proof makes use of martingale methods, it is pedagogically more suitable for students of financial mathematics than some other approaches. It is hoped that the end notes will also help to make the proof more accessible to this audience.
New Simple Method of Expansion of Iterated Ito Stochastic integrals of Multiplicity 2 Based on Expansion of the Brownian Motion Using Legendre Polynomials and Trigonometric Functions
Peter the Great Saint-Petersburg Polytechnic University（彼得大帝圣彼得堡理工大学）
In the present paper we show that the Levy-Ito representation of the infinitesimal generator $L$ for Levy processes $X_t$ can be written in a convolution-type form. Using the obtained convolution form we have constructed the quasi-potential operator $B$. We denote by $p(t,Δ)$ the probability that a sample of the process $X_t$ remains inside the domain $Δ$ for $0{\leq}τ{\leq}t$ (ruin problem). With the help of the operator $B$ we find a new formula for $p(t,Δ)$. This formula allows us to obtain long time behavior of $p(t,Δ)$.
In the present paper we study selfdecomposability of random fields, as defined directly rather than in terms of finite-dimensional distributions. The main tools in our analysis are the master Lévy measure and the associated Lévy-Itô representation. We give the dilation criterion for selfdecomposability analogous to the classical one. Next, we give necessary and sufficient conditions (in terms of the kernel functions) for a Volterra field driven by a Lévy basis to be selfdecomposable. In this context we also study the so-called Urbanik classes of random fields. We follow this with the study of …
Infinite Dimensional Ito Algebras of Quantum White Noise
给出了一般（无限维，非交换）Ito代数的一个简单的公理化刻画，并描述了这种代数的一个伪欧几里德基本表示。定义了Ito B ~*-代数的概念，推广了C ~*-代数的概念，使其包括量子布朗运动和量子Levy运动的Banach无穷维Ito代数，并刻画了真空和热量子噪声的B ~*-代数.证明了每一个Ito代数都可正则分解为量子Brownian（Wiener）代数与量子Levy（Poisson）代数的正交和.特别地，每个量子热噪声都是量子维纳噪声和量子泊松噪声的正交和，如经典情况下的Levy-Khinchin定理所述。
An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since…
We obtain a representation of an inhomogeneous Levy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Because the stochastic continuity is not assumed, our result generalizes the well known Levy-Ito representation for stochastic continuous processes with independent increments in Euclidean spaces and the extension to Lie groups.
After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman-Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations,this topological eq…
Let $Φ$ be a nuclear space and let $Φ'_β$ denote its strong dual. In this work we establish the one-to-one correspondence between infinitely divisible measures on $Φ'_β$ and Lévy processes taking values in $Φ'_β$. Moreover, we prove the Lévy-Itô decomposition, the Lévy-Khintchine formula and the existence of càdlàg versions for $Φ'_β$-valued Lévy processes. A characterization for Lévy measures on $Φ'_β$ is also established. Finally, we prove the Lévy-Khintchine formula for infinitely divisible measures on $Φ'_β$.
In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257--1283], we present an Itô multiple integral and a Stratonovich multiple integral with respect to a Lévy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the Itô multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu--Meyer formula that g…
In this paper we study first exit times from a bounded domain of a gradient dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.
In this work stochastic integration with respect to cylindrical Levy processes with weak second moments is introduced. It is well known that a deterministic Hilbert-Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Levy process by random Hilbert-Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic int…
Given an n-dimensional stochastic process X driven by P-Brownian motions and Poisson random measures, we seek the probability measure Q, with minimal relative entropy to P, such that the Q-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterise the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary var…
Ito's construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigro…
Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an…
We demonstrate the large deviation principle in the small noise limit for the mild solution of stochastic evolution equations with monotone nonlinearity. A recently developed method, weak convergent method, has been employed in studying the large deviations. we have used essentially the main result of Budhiraja et al., [4] which discloses the variational representation of exponential integrals w.r.t. the Lévy noise. An Itô-type inequality is a main tool in our proofs. Our framework covers a wide range of semilinear parabolic, hyperbolic and delay differential equations. We give some examples t…
We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial differential equations
This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Lévy basis) is studied in detail in terms of the free Lévy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosinski is developed. Finally a Lévy-Itô type decomposition for …
We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.
We show existence of a unique solution and a comparison theorem for a one-dimensional backward stochastic differential equation with jumps that emerge from a Lévy process. The considered generators obey a time-dependent extended monotonicity condition in the y-variable and have linear time-dependent growth. Within this setting, the results generalize those of Royer (2006), Yin and Mao (2008) and, in the $L^2$-case with linear growth, those of Kruse and Popier (2016). Moreover, we introduce an approximation technique: Given a BSDE driven by Brownian motion and Poisson random measure, we conside…
The purpose of this review article is to give an up to date account of the theory and application of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy-Khintchine formula and its relationship to the Levy-Ito decomposition. We shall also touch on more general topics such as excursion theory …
This is a survey of recent results on central and non-central limit theorems for quadratic functionals of stationary processes. The underlying processes are Gaussian, linear or Lévy-driven linear processes with memory, and are defined either in discrete or continuous time. We focus on limit theorems for Toeplitz and tapered Toeplitz type quadratic functionals of stationary processes with applications in parametric and nonparametric statistical estimation theory. We discuss questions concerning Toeplitz matrices and operators, Fejér-type singular integrals, and Lévy-Itô-type and Stratonovich-ty…
A comparison principle for stochastic integro-differential equations driven by Levy processes is proved. This result is obtained via an extension of an Ito formula from [11] for the square of the norm of the positive part of $L_2-$valued, continuous semimartingales, to the case of discontinuous semimartingales.
SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
For symmetric Lévy processes, if the local times exist, the Tanaka formula has already constructed via the techniques in the potential theory by Salminen and Yor (2007). In this paper, we study the Tanaka formula for arbitrary strictly stable processes with index $α\in (1,2)$ including spectrally positive and negative cases in a framework of Itô's stochastic calculus. Our approach to the existence of local times for such processes is different from Bertoin (1996).
A Lévy random medium, in a given space, is a random point process where the distances between points, a.k.a. targets, are long-tailed. Random walks visiting the targets of a Lévy random medium have been used to model many (physical, ecological, social) phenomena that exhibit superdiffusion as the result of interactions between an agent and a sparse, complex environment. In this note we consider the simplest non-trivial Lévy random medium, a sequence of points in the real line with i.i.d. long-tailed distances between consecutive targets. A popular example of a continuous-time random walk in th…
Semilinear stochastic evolution equations with Lévy noise and monotone nonlinear drift are considered. The existence and uniqueness of the mild solutions in $L^p$ for these equations is proved and a sufficient condition for exponential asymptotic stability of the solutions is derived. The main tool in our study is an Itô type inequality for the $p$th power of stochastic convolution integrals in Hilbert spaces.
We characterise the convergence of a certain class of discrete time Markov processes toward locally Feller processes in terms of convergence of associated operators. The theory of locally Feller processes is applied to Lévy-type processes in order to obtain convergence results on discrete and continuous time indexed processes, simulation methods and Euler schemes. We also apply the same theory to a slightly different situation, in order to get results of convergence of diffusions or random walks toward singular diffusions. As a consequence we deduce the convergence of random walks in random me…
The asymptotic solution of the inviscid Burgers equations with initial potential $ψ$ is closely related to the convex hull of the graph of $ψ$. In this paper, we study this convex hull, and more precisely its extremal points, if $ψ$ is a stochastic process. The times where those extremal points are reached, called extremal times, form a negligible set for Lévy processes, their integrated processes, and Itô processes. We examine more closely the case of a Lévy process with bounded variation. Its extremal points are almost surely countable, with accumulation only around the extremal values. Thes…
The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $σ$-finite measure on the space of $\{0,1\}\times\{0,1\}$-valued arrays. In discrete-time, this characterization amounts to a construction from an independent, identically distributed sequence of exchangeable random functions. In continuous-time, the behavior is enriched by a Lévy--Itô-type decomposition of the jump measure into mutually singular components that govern global, vertex-level, and edge-level dynamics. Furthermore, every such process almost surely projects to a Feller proces…
We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $σ$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a Lévy-Itô representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.
We study a random walk on a point process given by an ordered array of points $(ω_k, \, k \in \mathbb{Z})$ on the real line. The distances $ω_{k+1} - ω_k$ are i.i.d. random variables in the domain of attraction of a $β$-stable law, with $β\in (0,1) \cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $ω_k$ and $ω_\ell$ depend on $\ell-k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an $α$-stable law, with $α\in (0,1) \cup (1,2)$. Since the defining variables, for both the random walk and the po…
Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see \cite{Tank} for an overview), and more recently rough volatility models \cite{AlosLeon, Fukasawa}. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatil…
Ergodicity of stochastic functional differential equation with jumps and finite delay
School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）
The paper addresses one-dimensional transport in a Goupillaud medium (a layered medium in which the layer thickness is proportional to the propagation speed), as a prototypical case of wave propagation in random media. Suitable stochastic assumptions and limiting procedures lead to characteristic curves that are Lévy processes. Solutions corresponding to the discretely layered medium are shown to converge to limits as the thickness of the layers goes to zero. The probability distribution of the limiting characteristic curves is explicitly computed and exemplified when the underlying Lévy proce…
This work is concerned with forest and cumulant type expansions of general random variables on a filtered probability spaces. We establish a "broken exponential martingale" expansion that generalizes and unifies the exponentiation result of Al{ò}s, Gatheral, and Radoičić and the cumulant recursion formula of Lacoin, Rhodes, and Vargas. Specifically, we exhibit the two previous results as lower dimensional projections of the same generalized forest expansion, subsequently related by forest reordering. Our approach also leads to sharp integrability conditions for validity of the cumulant formula…
We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process $(y_t)_{t\geq 0}$ in the rough path topology. As an application, we establish weak convergence as $\varepsilon\to 0$ of the solution of the random ordinary differential equation (ODE) $\frac{d}{dt}x^\varepsilon_t=\frac{1}{\sqrt \varepsilon} f(x_t^\varepsilon,y_{\frac{t}{\varepsilon}})$ and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flo…
